Watch the mountain grow
Story: Each ball has to make lots of tiny left-right choices on the way down. Where will most of them land?
A Galton board is a triangle of pegs. You drop a ball from the top, and at every peg it bounces either left or right — each with 50% chance. At the bottom it lands in one of 13 buckets.
Drop enough balls and you get a bell curve: lots in the middle, few on the edges.
Start with 1 ball, then 10, then 100, then 1,000. The mountain shape gets easier to see as more balls fall.
Why the middle fills up
To reach the far left bucket (bucket 0) a ball must bounce left every single time — all 12 bounces. The chance of that is (½)^12 = about 0.024% — roughly 1 in 4,096.
To reach the middle bucket (bucket 6) the ball can bounce left and right in many different combinations. There are 924 ways to arrange 6 lefts and 6 rights across 12 bounces.
So the middle bucket has many different paths leading to it. The edge buckets have almost no choice at all.
Edge bucket: almost one exact path.
Middle bucket: lots of possible paths.
That is why the middle gets crowded. The middle has many roads leading to it.
This shape appears in lots of places
The same bell-shaped pattern appears in:
- Heights of people in your school
- Test scores across your class
- Measurement errors in any experiment
- Raindrop sizes in a storm
Whenever many tiny random choices pile up, the middle tends to win.
One important probability rule
All the bucket probabilities together must add up to 1, or 100%.
For our board with 12 rows and 13 buckets, all 13 probabilities add to 1.0000000000 — essentially exactly 1.
Quiz time
In a Galton board with 12 rows, how many buckets are there at the bottom?
With 12 rows of pegs, a ball can bounce right 0, 1, 2, … up to 12 times. That gives 13 possible landing spots.
Which bucket gets the MOST balls in a fair Galton board?
The middle bucket wins because there are many different combinations of left/right bounces that end up there. The edge buckets only have one path each (all-left or all-right).
What happens when you drop 1 000 balls into the Galton board?
With lots of balls, the shape becomes steadier and smoother. Big batches make the pattern easier to see.
In a Galton board with 12 rows, the ball bounces 12 times. If the board is perfectly fair (50-50 at each peg), which bucket number does it land in most often? (Count from 0 on the left.)
Bucket 6 — the exact middle. With 12 bounces and a fair 50-50 chance each time, the average number of 'right' bounces is exactly 6. The middle bucket wins most often.